Simply-connected $4$-manifolds with a given boundary

Abstract:

Let $M$ be a closed, oriented, connected $3$-manifold. For each bilinear, symmetric pairing $({{\mathbf {Z}}^n}, L)$, our goal is to calculate the set ${\mathcal {V}_L}(M)$ of all oriented homeomorphism types of compact, $1$-connected, oriented $4$-manifolds with boundary $M$ and intersection pairing isomorphic to $({{\mathbf {Z}}^n}, L)$. For each pair $({{\mathbf {Z}}^n}, L)$ which presents ${H_ \ast }(M)$, we construct a double coset space $B_L^t(M)$ and a function $c_L^t:{\mathcal {V}_L}(M) \to B_L^t(M)$. The set $B_L^t(M)$ is the quotient of the group of all link-pairing preserving isomorphisms of the torsion subgroup of ${H_1}(M)$ by two naturally occuring subgroups. When $({{\mathbf {Z}}^n}, L)$ is an even pairing, we construct another double coset space ${\hat B_L}(M)$, a function ${\hat c_L}:{\mathcal {V}_L}(M) \to {\hat B_L}(M)$ and a projection ${p_2}:{\hat B_L}(M) \to B_L^t(M)$ such that ${p_2} \cdot {\hat c_L} = c_L^t$. Our main result states that when $({{\mathbf {Z}}^n}, L)$ is even the function ${\hat c_L}$ is injective, as is the function $c_L^t \times \Delta :{\mathcal {V}_L}(M) \to B_L^t(M) \times {\mathbf {Z}}/2$ when $({{\mathbf {Z}}^n}, L)$ is odd. Here $\Delta$ is a Kirby-Siebenmann obstruction to smoothing. It follows that the sets ${\mathcal {V}_L}(M)$ are finite and of an order bounded above by a constant depending only on ${H_1}(M)$. We also show that when ${H_1}(M;{\mathbf {Q}}) \cong 0$ and $({{\mathbf {Z}}^n}, L)$ is even, $c_L^t = {p_2} \cdot {\hat c_L}$ is injective. It seems likely that via the functions $c_L^t \times \Delta$ and ${\hat c_L}$, the sets $B_L^t(M) \times {\mathbf {Z}}/2$ and ${\hat B_L}(M)$ calculate ${\mathcal {V}_L}(M)$ when $({{\mathbf {Z}}^n}, L)$ is respectively odd and even. We verify this in several cases, most notably when ${H_1}(M)$ is free abelian. The results above are based on a theorem which gives necessary and sufficient conditions for the existence of a homeomorphism between two $1$-connected $4$-manifolds extending a given homeomorphism of their boundaries. The theory developed is then applied to show that there is an $m > 0$, depending only on ${H_1}(M)$, such that for any self-homeomorphism $f$ of $M$, ${f^m}$ extends to a self-homeomorphism of any $1$-connected, compact $4$-manifold with boundary $M$.